The Anti-Magic Square Project

This web site documents my 1999 summer research on a combinatorial design called the Anti-Magic square. Anti-Magic Squares are a variation on the heavily studied and well-understood magic square. In contrast, very little seems to be known about AMSs. These pages describe what was previously known about the structure and history of the AMS and also detail new discoverys regarding their enumeration and construction.

Thanks for your patience and understanding, as this page is still under construction!

What is an Anti-Magic Square?

An Anti-Magic Square (AMS) is an arrangement of the numbers 1 to n2 in a square matrix such that the row, column, and diagonal sums form a sequence of consecutive integers. 1

The arrangement to the left is Anti-Magic because sorting the sums (numbers in black on the border) yields the sequence: 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269

This is an example of an AMS(8), or Anti-Magic Square of order 8, which comes from Madachy's Mathematical Recreations


Given this definition, this research project aims to answer some of the following questions:

And of course, we aim to have fun doing it! After all this is classified as "Recreational Math!"

Table of Contents

A present day Anti-Magic analogue to Durer's 1514 Magic Square

1. There has been some confusion about this definintion. Some people only require that the row, column and diagonal sums of a square be different to be called Anti-Magic. Such a design is actually called a heterosquare. A true AMS is a very special type of heterosquare.
Composed by John Cormie July 1999

Huge thanks go to my supervisor for the summer, Václav Linek

A research project funded by NSERC and The University of Winnipeg

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